How do i Snap an vertex just where it touches and edge following it´s own axis?
Blender has very limited precision modelling or CAD-like features, this is not natively supported at the moment.

You may look into third party addons to help you achieve this, some are commercial some are free:


You can try Blender TinyCAD addon (which I believe is shipped with Blender by default) that has a few intersection options. It doesn't directly provide snapping options, but can extend an existing edge until it reaches another one.
If you don't mind spending a little money, there is a payed addon Snap Utilities that provides a Sketchup-like workflow, with a pencil tool. (I am not in any way affiliated or associated with the addon creator, although I did purchase it).
There is also a free version of the above Snap Utilities, with some limitations in functionality